Calculus Examples

$e^{x} (x^{2} - 2 x + 2)$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = x^{2} - 2 x + 2$ .

$e^{x} \frac{d}{d x} [x^{2} - 2 x + 2] + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2

Differentiate.

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Step 2.1

By the Sum Rule, the derivative of $x^{2} - 2 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [2]$ .

$e^{x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [2]) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$e^{x} (2 x + \frac{d}{d x} [- 2 x] + \frac{d}{d x} [2]) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2.3

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2 x$ with respect to $x$ is $- 2 \frac{d}{d x} [x]$ .

$e^{x} (2 x - 2 \frac{d}{d x} [x] + \frac{d}{d x} [2]) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{x} (2 x - 2 \cdot 1 + \frac{d}{d x} [2]) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2.5

Multiply $- 2$ by $1$ .

$e^{x} (2 x - 2 + \frac{d}{d x} [2]) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2.6

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$e^{x} (2 x - 2 + 0) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 2.7

Add $2 x - 2$ and $0$ .

$e^{x} (2 x - 2) + (x^{2} - 2 x + 2) \frac{d}{d x} [e^{x}]$

Step 3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} (2 x - 2) + (x^{2} - 2 x + 2) e^{x}$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$e^{x} (2 x) + e^{x} \cdot - 2 + (x^{2} - 2 x + 2) e^{x}$

Step 4.2

Apply the distributive property.

$e^{x} (2 x) + e^{x} \cdot - 2 + x^{2} e^{x} - 2 x e^{x} + 2 e^{x}$

Step 4.3

Combine terms.

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Step 4.3.1

Move $- 2$ to the left of $e^{x}$ .

$e^{x} (2 x) - 2 \cdot e^{x} + x^{2} e^{x} - 2 x e^{x} + 2 e^{x}$

Step 4.3.2

Subtract $2 x e^{x}$ from $e^{x} (2 x)$ .

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Step 4.3.2.1

Move $e^{x}$ .

$2 x e^{x} - 2 x e^{x} - 2 e^{x} + x^{2} e^{x} + 2 e^{x}$

Step 4.3.2.2

Subtract $2 x e^{x}$ from $2 x e^{x}$ .

$0 - 2 e^{x} + x^{2} e^{x} + 2 e^{x}$

Step 4.3.3

Subtract $2 e^{x}$ from $0$ .

$- 2 e^{x} + x^{2} e^{x} + 2 e^{x}$

Step 4.3.4

Add $- 2 e^{x}$ and $2 e^{x}$ .

$x^{2} e^{x} + 0$

Step 4.3.5

Add $x^{2} e^{x}$ and $0$ .

$x^{2} e^{x}$

Step 4.4

Reorder the factors of $x^{2} e^{x}$ .

$e^{x} x^{2}$

Step 4.5

Reorder factors in $e^{x} x^{2}$ .

$x^{2} e^{x}$